Use with the (group-theory) tag. The tag "finite-groups" refers to questions asked in the field of Group Theory which, in particular, focus on the groups of finite order.

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Operation with normal subgroup

I am working on a problem on finite group theory, and would like asking a question on the correct operation of normal subgroup. Suppose that $H$ is normal subgroup of $G$ and the factor group $G/H$ ...
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16 views

If G is a p-group, the number of nonnormal subgroup is a multiple of p? [duplicate]

I want to show that If $G$ is a finite $p$-group, then the number of nonnormal subgroups of $G$ is a multiple of $p$. I think I need to consider a conjugation action. But then?
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1answer
48 views

Find number of Sylow $3$ and Sylow $5$ subgroups of a simple Group, $G$ of order $60$

So I wanted to check if what I did was correct. I'm not sure if it is and if so what would the correct way to go about this be? So firstly $|G| = 2^2*3*5$. This confirms there is Sylow 3 - ...
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2answers
67 views

Kernel of a homomorphism of a group algebra

Excuse me, this is stupid, but I have a short circuit in my head, I can't understand the situation. Let $k$ be a field, $G$ a finite group, $kG$ the corresponding group algebra and $\delta:G\to kG$ ...
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2answers
43 views

To make $(K_4,+)$ ( the Klein-4 group ) a ring

How can we define an operation $.$ such that the Klein-4 group $(K_4,+)$ becomes a ring $(K_4,+,.)$ ?
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26 views

Ways of writing an $n$-cycle as product of a $2$-cycle and $n-1$ cycle.

We know that any $n$ cycle can be written as a product of a $2$-cycle and an $n-1$ cycle; but this decomposition is not unique: $(123)=(12)(23)$ and $(123)=(23)(31)$ [product taken from right to left ...
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1answer
28 views

A sub-theorem inside “Prove that The group $A_4$ has no subgroup of order $6$”.

Proposition: The group $A_4$ has no subgroup of order $6$. Proof: Suppose we have some $H$ such that $|H|=6$, thus $[A_4 : H ] = 2$. Thus there are only two cosets of $H$ in $A_4$ . Inasmuch as one ...
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49 views

Sylow 5-subgroups of groups of order $2^n5^m$ are normal

My textbook says: Show that a group of order $2^n5^m, m, n \ge 1$ has a normal 5-Sylow subgroup. I've been banging my head against this problem for days with no success, how can I prove this?
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26 views

Number of Sylow $2$-subgroups in dihedral group $D_{20}$

By Sylow's theorem I know that the number of Sylow $2$-subgroups in the symmetry group of a regular $10$-gon $D_{20}$ is either $1$ or $5$. How do I exclude the possibility $1$?
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79 views

Why do Sylow $p$-groups in finite simple group have trivial intersection?

I'm looking for an argument for the fact that two Sylow $p$-subgroups in a finite simple group have trivial intersection. In the case that the order of the Sylow subgroups is the prime $p$ it is easy, ...
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1answer
58 views

What is set builder of $\langle H, K \rangle$?

I am looking left and right for a lemma to solve a problem on solvable group here, and I think I have found one under commutator group: For any two subgroups $H, K$ of $G$, the $[H, K]$ is a ...
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58 views

What is the composition series of $\mathbb Z_7$ x $\mathbb Z_{12}$

So I get the answer as follows (which is correct I believe): {$0$} x {$0$} $\vartriangleleft$ $\mathbb Z_7$ x {$0$} $\vartriangleleft$ $\mathbb Z_7$ x <$4$> $\vartriangleleft$ $\mathbb Z_7$ x ...
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20 views

Finite Groups of Isometries of Euclidean spaces

Six years before, I had a collection of articles on finite groups of rotations of sphere, from Monthly, Gazette, Intelligencer. But I lost these articles. Since I am teaching this topic, I would like ...
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23 views

group cohomology of permutation groups

Let $\Sigma_k$ be the permutation group of order $k$. Let $F$ be a field. What is the cohomology $$ H^*(\Sigma_k;F)=H^*(K(\Sigma_k,1);F)=H^*(B\Sigma_k;F)? $$ For $F=\mathbb{Z}/p\mathbb{Z}$ for prime ...
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2answers
126 views

Reconciling Different Definitions of Solvable Group

Looks like my class note defines solvable group differently from others: A finite group $G$ is called solvable if $H' \neq H$ for each subgroup $H$ of $G$ different from $\{1\}$, where $H'$ is ...
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1answer
44 views

Basic group algebra exercise

I'm working on this abstract algebra problem that was given as homework to me, I'm in my first year of university and have a semester's background in basic abstract algebra. Here is the problem: Let ...
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56 views

Kernel and image of a homomorphism $SL(2,5)\to S_5$

Since $SL(2,5)$ has a subgroup of index $5$, I can use the left coset action to define a homomorphism between $SL(2,5)$ and $S_5$. How can I find the kernel and the image of this homomorphism? ...
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58 views

Capable group of order 32

A group that can be written as $G/Z(G)$ for some group $G$ is called capable. Can someone list the capable groups of order 32?
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85 views

Elementary Applications of Cayley's Theorem in Group Theory

The Cayley's theorem says that every group $G$ is a subgroup of some symmetric group. More precisely, if $G$ is a group of order $n$, then $G$ is a subgroup of $S_n$. In the course on group theory, ...
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3answers
144 views

Group acting on its subsets

Let $ G $ be a group with $ |G|=mp^\alpha $ where $ \alpha\geq1 $ and p is prime integer with $p \nmid m$. Then denote the set of subsets of G, having $p^\alpha$ size, with $X$. Then with the action ...
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53 views

Showing that $\{0, 1\}$ is a group under addition modulo $2$

I'm considering a set $ G = \{0,1\}$ under addition modulo 2. I.e. $$ a*b = a + b\bmod 2, \quad \quad \forall \ a,b \in G. $$ I am able to show that there exists an identity element, $0$. Showing ...
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37 views

on the characters of the normal subgroup and its quotient

I read character theory recently and thought about the following proposition, but I do not know is this true or false: Let $G $ be a finite group such that two distinct primes $ p $ and $ q $ ...
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Determine all the normal subgroups of $S_4$ [duplicate]

Previous part: Give a representative of each conjugacy class of $G$ - $e,(1,2),(1,2,3),(1,2,3,4),(1,2)(3,4)$ Calculate the size of each conjugacy class of $G$. For: $e-1$, $(1,2)-6$, ...
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1answer
57 views

Finite Group is Subgroup of Its Radical's Automorphism

I am still working on this problem on radical of finite group: Assume that $R(G)$ is simple and not commutative, show that $G$ is a subgroup of $Aut(R(G))$. I have managed to parse the problem ...
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1answer
22 views

Find a composition series of $D_8$

Answer: Let $D_8=\langle a,b\rangle=\{e,a,a^2,a^3,b,ba,ba^2,ba^3\}$, where $a^4 = b^2 = e$ and $ab = ba^{-1}$ as usual. Then $$\{e\}\lhd\langle a^2\rangle\lhd\langle a\rangle\lhd D_8$$ or ...
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2answers
67 views

No group of order $400$ is simple - clarification

I was reading through a proof that no group of order $400$ is simple which can be found here: http://math.stackexchange.com/a/79644/169389 Here is an outline for a solution. First of all, ...
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28 views

Reference request: groups of order $p^4$.

I am looking for a textbook or a paper which include the classification of groups of order $p^4$ ($p$ is prime) using generators and relations. In particular I like to understand which group $G$ ...
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1answer
53 views

Calculating the second cohomology group for trivial group action

Let $G$ be a finite group acting trivially on $\mathbb{R}^*$. How can I compute $H^2(G,\mathbb{R}^*)$? It seems that direct calculations are somewhat hopeless, but the answer should be simple anyway.
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1answer
69 views

Simple Groups of Finite Order

Prove that there is NO simple group of order $n$ for each the following integers: $n=88, n=96, n=132.$ I am supposed to solve this using Sylows theorems somehow. Lets start with $n = 88$ and say ...
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1answer
83 views

What is a generator of a finite cyclic group? (General)

I have asked a few questions about this but I am still confused. So, in general, what is a generator of a finite cyclic group and how is it found? I have seen in books and my notes a lot of ...
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67 views

A group of order $2^67$

Let $G$ be a finite group of order $2^6\times 7$. Also $G$ has at least $5$ irreducible characters of degree $7$. I want to prove that with this condition there exists no character of even degree in ...
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1answer
42 views

Proof-verification group order 48 not simple

Is the following proof correct? Let $G$ be a group of order 48. Let's prove it is not simple. $\lvert G\rvert=48=2^4\cdot3$. By Sylow's Theorem, $n_3\in\{1,4,16\},\:n_2\in\{1,3\}$. $n_3=1$ or ...
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61 views

$\lvert G\rvert=24$ not simple by a counting arguments [duplicate]

I want to prove that $\lvert G\rvert=24$, then it has a non-trivial normal subgroup. Here is my attempt: $n_2$: number of 2-Sylow subgroup; $n_3$: number of 3-Sylow subgroup. ...
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1answer
98 views

$|G|=24$ prove that $G$ is not simple

Let $G$ be a group of order 24, and we shall assume there there exist a non-normal 2-sylow group in $G$. i want to show that it's not simple. first i have showed that there are exactly three 2-sylow ...
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2answers
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There is no simple group of order $144$

There is no simple group of order $144$ I have a question to the proof of the statement above (from the book J. Gallian, Contemporary abstract algebra), it is about the index theorem, so I give ...
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3answers
59 views

If $G$ be a finite group prove that $G=AB$

Let $G$ be a finite group. Suppose that $A$ and $B$ are to subsets of $G$. If $|G|<|A|+|B|$ prove that $$G=AB$$
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32 views

Hom$_{Set}(G \times H,R) \cong $Hom$_{Set}(G,R)\otimes $ Hom$_{Set}(H,R)$?

If $G$ and $H$ are finite groups and $R$ is a ring does Hom$_{Set}(G \times H,R) \cong $Hom$_{Set}(G,R)\otimes $ Hom$_{Set}(H,R)$ if Hom sets have the usual R-algebra structure?
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Frattini subgroup of a $p$- group of order $p^4$

Let $p$ be an odd prime and $G$ be a finite non-abelian $p$-group of order $p^4$ with the following presentation: $$\langle a, b, c, d\mid a^p=b^p=c^p=d^p=1, c^d=cb, b^d=ba, ...
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1answer
38 views

A group of order $12$ either has a normal $ 3$-Sylow subgroup or is isomorphic to $A_4$

Let $G$ be a group of order $12.$ Prove that either $G$ has a normal $ 3$-Sylow subgroup or $ G$ is isomorphic to $A_4$. I know that $|G|=12=2^23$ and that either $n_3=1$ and there is a $3$-sylow ...
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1answer
26 views

Element commuting with normal subgroup of $p$-group

Let $p$ be a prime. Suppose $N\triangleleft G$ where $|G| = p^n$ ($n>2$) and $|N| = p$. Prove there exists $g\not\in N$ such that $gn = ng$ for all $n\in N$. I am supposed to prove this ...
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1answer
36 views

For a special group G described below, the order of G can only be $p_1p_2…p_k$ where $p_i$ s are prime.

I am just learning generators and relations basics. A clarification Let $G$ be a finite group in which every element(non-zero) is a possible candidate to be in the minimal generating set $S$ of the ...
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36 views

Find the number of subgroups of $\mathbb{Z}/(5)\times \mathbb{Z}/(5)$

Find the number of subgroups of $\mathbb{Z}/(5)\times \mathbb{Z}/(5)$ including trivial subgroups. My Work: If we consider $\mathbb{Z}/(5)$ then the only subgroups are trivial subgroups. But how ...
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How we can conclude that $p\nmid \sum_{x\in H}|x^G|$ in a group with some elements of order $2p$?

Let $G$ be a finite group such that has some elements of order $2p$, where $p$ is an odd prime. Let $H$ be the set of all elements of order $2p$ in $G$. We can show $G$ acts on $H$ by conjugation. So ...
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1answer
36 views

Are the primitive groups linearly primitive?

A transitive permutation group $G \subset S_n$ is primitive if $G_1 \subset G$ is a maximal subgroup. A finite group $G$ is linearly primitive if it has a faithful complex irreducible representation. ...
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1answer
77 views

Subgroups of Semidirect Product of the elementary abelian group of order 8 by $S_3$

What are the subgroups of the semidirect product of the elementary abelian group of order 8 by $S_3$? This is the group $(\mathbb{Z}_2\times \mathbb{Z}_2\times \mathbb{Z}_2)\rtimes S_3$ of order 48; ...
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Which inclusions of finite groups are relatively linearly primitive?

This post is a sequel of: Which finite groups have faithful complex irreducible representations? A finite group $G$ is linearly primitive if it has a faithful complex irreducible representation. ...
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70 views

A group of order $pqr$ (primes $p > q > r$) has a subgroup of order $qr$ [duplicate]

I've done most of the following problem, but I can't seem to get part (d). Let $G$ be a group of order $pqr$ for primes $p > q > r$. By a counting argument one can see that there is ...
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1answer
36 views

$G$ is characteristically simple $\iff$ there is simple $T$ such that $G \cong T\times T \times \cdots \times T$

Let $G$ be a characteristically simple finite group, i.e. it has no nontrivial characteristic subgroups. Prove there is some simple group $T$ such that $G \cong T \times T \times \cdots \times T$. ...
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38 views

The finite groups with an irreducible faithful complex representation

All the groups below are supposed finite, and their representations, complex. An abelian group admits an irreducible faithful representation iff it is cyclic. A group has all its non-trivial ...
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40 views

What is the most mundane & intuitive meaning of the radical of a finite group?

I am asking this question because I am trying to solve this problem from a class note: Assume that $R(G)$ is simple and not commutative, show that $G$ is a subgroup of $Aut(R(G)).$ The note's ...